:
In this talk, I will talk about the
properties and the classification of embeddings of homogeneous spaces,
especially the case of affine normal embeddings of reductive groups. We might
guess that as in the case of toric varieties, some specific subset of one-parameter
subgroups may contribute to the classification of affine embeddings of general
reductive group.

To check this, we review the theory
of affine normal SL(2)-embeddings, and prove that the classification cannot be
solved entirely based on one-parameter subgroups. Also, I will also give
examples of GL(2)-embeddings which had not previously been constructed in
detail, which might be helpful in understanding the general classification of
affine normal G-embeddings.

One interesting properties of
SL(2)-embeddings and GL(2)-embeddings are that they are Mori dream spaces with
some general conditions. If time permitted, I will review how to describe the
geometry of SL(2)-equivariant flips, and try to do the same thing for GL(2)-equivariant flips.